Abstract
In this chapter, we extend the ideas of the model reference adaptive control (MRAC), developed for integer-order plants with integer-order adaptive laws, to the case of integer-order plants but with fractional-order adaptive laws. Two cases are analyzed in detail; the direct MRAC (DMRAC) and the combined MRAC (CMRAC). In both cases, boundedness of all the signals in the resultant adaptive scheme is theoretically proved and a discussion on the error, and parameter convergence is provided in each case. The study is performed for scalar first-order time-invariant plants, since extensions to the vector case are currently under investigation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aguila-Camacho, N., Duarte-Mermoud, M.A.: Fractional adaptive control for an automatic voltage regulator. ISA Trans. 52, 807–815 (2013)
Aguila-Camacho, N., Duarte-Mermoud, M.A.: Boundedness of the solutions for certain classes of fractional differential equations with applications to adaptive systems. ISA Trans. 60, 82–88 (2016)
Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19, 2951–2957 (2014)
Alikhanov, A.: A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equ. 46, 660–666 (2010)
Astrom, K.J., Bohlin, T.: Numerical Identification of Linear Dynamic Systems From Normal Operating Records, pp. 96–111. Springer, Boston (1966)
Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Heidelberg (2004)
Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., Castro-Linares, R.: Using general quadratic lyapunov functions to prove lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22, 650–659 (2015)
Duarte-Mermoud, M.A., Narendra, K.S.: Combined direct and indirect approach to adaptive control. IEEE Trans. Autom. Control 34, 1071–1075 (1989)
Duarte-Mermoud, N.A., Narendra, K.S.: A new approach to model reference adaptive control. Int. J. Adapt. Control Signal Process. 3(1), 53–73 (1989)
Gallegos, J.A., Duarte-Mermoud, M.A.: Boundedness and convergence on fractional order systems. J. Comput. Appl. Math. 296, 815–826 (2016)
Gallegos, J.A., Duarte-Mermoud, M.A.: Mixed order robust adaptive control. Submitt. J. Franklin Inst. (2017)
Gallegos, J.A., Duarte-Mermoud, M.A.: Robustness and convergence of fractional systems and their applications to adaptive systems. Submitted to Fractional Calculus and Applied Analysis (2017)
Gallegos, J.A., Duarte-Mermoud, M.A., Aguila-Camacho, N.: Smoothness and boundedness on system of mixed-order fractional differential equations. Technical Report 2016-1, Santiago, Chile: Department of Electrical Engineering, University of Chile (2016)
Goodwin, G.C., Ramadge, P.J., Caines, P.E.: Discrete time multivariable adaptive control. IEEE Trans. Autom. Control AC-25(3), 449–456 (1980)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Morse, A.S.: Global stability of parameter-adaptive control systems. IEEE Trans. Autom. Control AC-25(3), 433–439 (1980)
Narendra, K.S., Annaswamy, A.M.: Persistent excitation of adaptive systems. Int. J. Control 45, 127–160 (1987)
Narendra, K.S., Annaswamy, A.M.: Stable Adaptive Systems. Dover Publications Inc., Mineola (2005)
Narendra, K.S., Duarte-Mermoud, M.A.: Robust adaptive control using direct and indirect methods. In: A.A.C. Council (ed.) Proceedings of the 7th American Control Conference, vol. 3, pp. 2429–2433. Atlanta, Georgia, USA (1988)
Narendra, K.S., Lin, Y.H.: Stable discrete adaptive control. IEEE Trans. Autom. Control AC-25(3), 456–461 (1980)
Narendra, K.S., Lin, Y.H., Valavani, L.S.: Stable adaptive controller design part ii: Proof of stability. IEEE Trans. Autom. Control AC-25(3), 440–480 (1980)
Oustaloup, A.: La commande CRONE: commande robuste d’ordre non entier. Hermes, Paris (1991)
Sastry, S., Bodson, M.: Adaptive Control: Stability Convergence and Robustness. Prentice Hall, Upper Saddle River (1994)
Suárez, J.I., Vinagre, B.M., Chen, Y.Q.: A fractional adaptation scheme for lateral control of an AGV. J. Vibr. Control 14, 1499–1511 (2008)
Valério, D., Da Costa, J.S.: Ninteger: a non-integer control toolbox for matlab. In: IFAC (ed.) Fractional Derivatives and Applications. Bordeaux, France (2004)
Vinagre, B.M., Petrás, I., Podlubny, I., Chen, Y.Q.: Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control. Nonlinear Dyn. 29, 269–279 (2002)
Whitaker, H.P., Yamron, J., Kezer, A.: Design of model reference adaptive control systems for aircraft. Technical Report R-164, Instrumentation Laboratory, MIT, Cambridge, MA (1958)
Acknowledgements
The results reported in this chapter have been financed by CONICYT- Chile, under the Basal Financing Program FB0809 “Advanced Mining Technology Center”, FONDECYT Project 1150488, “Fractional Error Models in Adaptive Control and Applications”, and FONDECYT 3150007, “Postdoctoral Program 2015”.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A., Travieso-Torres, J.C. (2018). Fractional-Order Model Reference Adaptive Controllers for First-Order Integer Plants. In: Clempner, J., Yu, W. (eds) New Perspectives and Applications of Modern Control Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-62464-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-62464-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62463-1
Online ISBN: 978-3-319-62464-8
eBook Packages: EngineeringEngineering (R0)